3.158 \(\int \frac{a+b x+c x^2}{x^4 \sqrt{-1+d x} \sqrt{1+d x}} \, dx\)

Optimal. Leaf size=116 \[ \frac{\sqrt{d x-1} \sqrt{d x+1} \left (2 a d^2+3 c\right )}{3 x}+\frac{a \sqrt{d x-1} \sqrt{d x+1}}{3 x^3}+\frac{1}{2} b d^2 \tan ^{-1}\left (\sqrt{d x-1} \sqrt{d x+1}\right )+\frac{b \sqrt{d x-1} \sqrt{d x+1}}{2 x^2} \]

[Out]

(a*Sqrt[-1 + d*x]*Sqrt[1 + d*x])/(3*x^3) + (b*Sqrt[-1 + d*x]*Sqrt[1 + d*x])/(2*x
^2) + ((3*c + 2*a*d^2)*Sqrt[-1 + d*x]*Sqrt[1 + d*x])/(3*x) + (b*d^2*ArcTan[Sqrt[
-1 + d*x]*Sqrt[1 + d*x]])/2

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Rubi [A]  time = 0.456962, antiderivative size = 171, normalized size of antiderivative = 1.47, number of steps used = 7, number of rules used = 7, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.219 \[ -\frac{\left (1-d^2 x^2\right ) \left (2 a d^2+3 c\right )}{3 x \sqrt{d x-1} \sqrt{d x+1}}-\frac{a \left (1-d^2 x^2\right )}{3 x^3 \sqrt{d x-1} \sqrt{d x+1}}-\frac{b \left (1-d^2 x^2\right )}{2 x^2 \sqrt{d x-1} \sqrt{d x+1}}+\frac{b d^2 \sqrt{d^2 x^2-1} \tan ^{-1}\left (\sqrt{d^2 x^2-1}\right )}{2 \sqrt{d x-1} \sqrt{d x+1}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x + c*x^2)/(x^4*Sqrt[-1 + d*x]*Sqrt[1 + d*x]),x]

[Out]

-(a*(1 - d^2*x^2))/(3*x^3*Sqrt[-1 + d*x]*Sqrt[1 + d*x]) - (b*(1 - d^2*x^2))/(2*x
^2*Sqrt[-1 + d*x]*Sqrt[1 + d*x]) - ((3*c + 2*a*d^2)*(1 - d^2*x^2))/(3*x*Sqrt[-1
+ d*x]*Sqrt[1 + d*x]) + (b*d^2*Sqrt[-1 + d^2*x^2]*ArcTan[Sqrt[-1 + d^2*x^2]])/(2
*Sqrt[-1 + d*x]*Sqrt[1 + d*x])

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Rubi in Sympy [A]  time = 25.0132, size = 119, normalized size = 1.03 \[ \frac{2 a d^{2} \sqrt{d x - 1} \sqrt{d x + 1}}{3 x} + \frac{a \sqrt{d x - 1} \sqrt{d x + 1}}{3 x^{3}} + \frac{b d^{2} \operatorname{atan}{\left (\sqrt{d x - 1} \sqrt{d x + 1} \right )}}{2} + \frac{b \sqrt{d x - 1} \sqrt{d x + 1}}{2 x^{2}} + \frac{c \sqrt{d x - 1} \sqrt{d x + 1}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x+a)/x**4/(d*x-1)**(1/2)/(d*x+1)**(1/2),x)

[Out]

2*a*d**2*sqrt(d*x - 1)*sqrt(d*x + 1)/(3*x) + a*sqrt(d*x - 1)*sqrt(d*x + 1)/(3*x*
*3) + b*d**2*atan(sqrt(d*x - 1)*sqrt(d*x + 1))/2 + b*sqrt(d*x - 1)*sqrt(d*x + 1)
/(2*x**2) + c*sqrt(d*x - 1)*sqrt(d*x + 1)/x

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Mathematica [A]  time = 0.144746, size = 75, normalized size = 0.65 \[ \frac{1}{6} \left (\frac{\sqrt{d x-1} \sqrt{d x+1} \left (a \left (4 d^2 x^2+2\right )+3 x (b+2 c x)\right )}{x^3}-3 b d^2 \tan ^{-1}\left (\frac{1}{\sqrt{d x-1} \sqrt{d x+1}}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x + c*x^2)/(x^4*Sqrt[-1 + d*x]*Sqrt[1 + d*x]),x]

[Out]

((Sqrt[-1 + d*x]*Sqrt[1 + d*x]*(3*x*(b + 2*c*x) + a*(2 + 4*d^2*x^2)))/x^3 - 3*b*
d^2*ArcTan[1/(Sqrt[-1 + d*x]*Sqrt[1 + d*x])])/6

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Maple [C]  time = 0.03, size = 123, normalized size = 1.1 \[ -{\frac{ \left ({\it csgn} \left ( d \right ) \right ) ^{2}}{6\,{x}^{3}}\sqrt{dx-1}\sqrt{dx+1} \left ( 3\,\arctan \left ({\frac{1}{\sqrt{{d}^{2}{x}^{2}-1}}} \right ){x}^{3}b{d}^{2}-4\,\sqrt{{d}^{2}{x}^{2}-1}{x}^{2}a{d}^{2}-6\,\sqrt{{d}^{2}{x}^{2}-1}{x}^{2}c-3\,bx\sqrt{{d}^{2}{x}^{2}-1}-2\,a\sqrt{{d}^{2}{x}^{2}-1} \right ){\frac{1}{\sqrt{{d}^{2}{x}^{2}-1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x+a)/x^4/(d*x-1)^(1/2)/(d*x+1)^(1/2),x)

[Out]

-1/6*(d*x-1)^(1/2)*(d*x+1)^(1/2)*csgn(d)^2*(3*arctan(1/(d^2*x^2-1)^(1/2))*x^3*b*
d^2-4*(d^2*x^2-1)^(1/2)*x^2*a*d^2-6*(d^2*x^2-1)^(1/2)*x^2*c-3*b*x*(d^2*x^2-1)^(1
/2)-2*a*(d^2*x^2-1)^(1/2))/(d^2*x^2-1)^(1/2)/x^3

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Maxima [A]  time = 1.50168, size = 119, normalized size = 1.03 \[ -\frac{1}{2} \, b d^{2} \arcsin \left (\frac{1}{\sqrt{d^{2}}{\left | x \right |}}\right ) + \frac{2 \, \sqrt{d^{2} x^{2} - 1} a d^{2}}{3 \, x} + \frac{\sqrt{d^{2} x^{2} - 1} c}{x} + \frac{\sqrt{d^{2} x^{2} - 1} b}{2 \, x^{2}} + \frac{\sqrt{d^{2} x^{2} - 1} a}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)/(sqrt(d*x + 1)*sqrt(d*x - 1)*x^4),x, algorithm="maxima")

[Out]

-1/2*b*d^2*arcsin(1/(sqrt(d^2)*abs(x))) + 2/3*sqrt(d^2*x^2 - 1)*a*d^2/x + sqrt(d
^2*x^2 - 1)*c/x + 1/2*sqrt(d^2*x^2 - 1)*b/x^2 + 1/3*sqrt(d^2*x^2 - 1)*a/x^3

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Fricas [A]  time = 0.229265, size = 297, normalized size = 2.56 \[ -\frac{12 \, b d^{4} x^{5} - 12 \, c d^{2} x^{4} - 15 \, b d^{2} x^{3} - 6 \,{\left (a d^{2} - c\right )} x^{2} - 3 \,{\left (4 \, b d^{3} x^{4} - 4 \, c d x^{3} - 3 \, b d x^{2} - 2 \, a d x\right )} \sqrt{d x + 1} \sqrt{d x - 1} + 3 \, b x - 6 \,{\left (4 \, b d^{5} x^{6} - 3 \, b d^{3} x^{4} -{\left (4 \, b d^{4} x^{5} - b d^{2} x^{3}\right )} \sqrt{d x + 1} \sqrt{d x - 1}\right )} \arctan \left (-d x + \sqrt{d x + 1} \sqrt{d x - 1}\right ) + 2 \, a}{6 \,{\left (4 \, d^{3} x^{6} - 3 \, d x^{4} -{\left (4 \, d^{2} x^{5} - x^{3}\right )} \sqrt{d x + 1} \sqrt{d x - 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)/(sqrt(d*x + 1)*sqrt(d*x - 1)*x^4),x, algorithm="fricas")

[Out]

-1/6*(12*b*d^4*x^5 - 12*c*d^2*x^4 - 15*b*d^2*x^3 - 6*(a*d^2 - c)*x^2 - 3*(4*b*d^
3*x^4 - 4*c*d*x^3 - 3*b*d*x^2 - 2*a*d*x)*sqrt(d*x + 1)*sqrt(d*x - 1) + 3*b*x - 6
*(4*b*d^5*x^6 - 3*b*d^3*x^4 - (4*b*d^4*x^5 - b*d^2*x^3)*sqrt(d*x + 1)*sqrt(d*x -
1))*arctan(-d*x + sqrt(d*x + 1)*sqrt(d*x - 1)) + 2*a)/(4*d^3*x^6 - 3*d*x^4 - (4
*d^2*x^5 - x^3)*sqrt(d*x + 1)*sqrt(d*x - 1))

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Sympy [A]  time = 156.309, size = 219, normalized size = 1.89 \[ - \frac{a d^{3}{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{9}{4}, \frac{11}{4}, 1 & \frac{5}{2}, \frac{5}{2}, 3 \\2, \frac{9}{4}, \frac{5}{2}, \frac{11}{4}, 3 & 0 \end{matrix} \middle |{\frac{1}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} - \frac{i a d^{3}{G_{6, 6}^{2, 6}\left (\begin{matrix} \frac{3}{2}, \frac{7}{4}, 2, \frac{9}{4}, \frac{5}{2}, 1 & \\\frac{7}{4}, \frac{9}{4} & \frac{3}{2}, 2, 2, 0 \end{matrix} \middle |{\frac{e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} - \frac{b d^{2}{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{7}{4}, \frac{9}{4}, 1 & 2, 2, \frac{5}{2} \\\frac{3}{2}, \frac{7}{4}, 2, \frac{9}{4}, \frac{5}{2} & 0 \end{matrix} \middle |{\frac{1}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} + \frac{i b d^{2}{G_{6, 6}^{2, 6}\left (\begin{matrix} 1, \frac{5}{4}, \frac{3}{2}, \frac{7}{4}, 2, 1 & \\\frac{5}{4}, \frac{7}{4} & 1, \frac{3}{2}, \frac{3}{2}, 0 \end{matrix} \middle |{\frac{e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} - \frac{c d{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{5}{4}, \frac{7}{4}, 1 & \frac{3}{2}, \frac{3}{2}, 2 \\1, \frac{5}{4}, \frac{3}{2}, \frac{7}{4}, 2 & 0 \end{matrix} \middle |{\frac{1}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} - \frac{i c d{G_{6, 6}^{2, 6}\left (\begin{matrix} \frac{1}{2}, \frac{3}{4}, 1, \frac{5}{4}, \frac{3}{2}, 1 & \\\frac{3}{4}, \frac{5}{4} & \frac{1}{2}, 1, 1, 0 \end{matrix} \middle |{\frac{e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+b*x+a)/x**4/(d*x-1)**(1/2)/(d*x+1)**(1/2),x)

[Out]

-a*d**3*meijerg(((9/4, 11/4, 1), (5/2, 5/2, 3)), ((2, 9/4, 5/2, 11/4, 3), (0,)),
1/(d**2*x**2))/(4*pi**(3/2)) - I*a*d**3*meijerg(((3/2, 7/4, 2, 9/4, 5/2, 1), ()
), ((7/4, 9/4), (3/2, 2, 2, 0)), exp_polar(2*I*pi)/(d**2*x**2))/(4*pi**(3/2)) -
b*d**2*meijerg(((7/4, 9/4, 1), (2, 2, 5/2)), ((3/2, 7/4, 2, 9/4, 5/2), (0,)), 1/
(d**2*x**2))/(4*pi**(3/2)) + I*b*d**2*meijerg(((1, 5/4, 3/2, 7/4, 2, 1), ()), ((
5/4, 7/4), (1, 3/2, 3/2, 0)), exp_polar(2*I*pi)/(d**2*x**2))/(4*pi**(3/2)) - c*d
*meijerg(((5/4, 7/4, 1), (3/2, 3/2, 2)), ((1, 5/4, 3/2, 7/4, 2), (0,)), 1/(d**2*
x**2))/(4*pi**(3/2)) - I*c*d*meijerg(((1/2, 3/4, 1, 5/4, 3/2, 1), ()), ((3/4, 5/
4), (1/2, 1, 1, 0)), exp_polar(2*I*pi)/(d**2*x**2))/(4*pi**(3/2))

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GIAC/XCAS [A]  time = 0.228705, size = 266, normalized size = 2.29 \[ -\frac{3 \, b d^{3} \arctan \left (\frac{1}{2} \,{\left (\sqrt{d x + 1} - \sqrt{d x - 1}\right )}^{2}\right ) + \frac{2 \,{\left (3 \, b d^{3}{\left (\sqrt{d x + 1} - \sqrt{d x - 1}\right )}^{10} - 12 \, c d^{2}{\left (\sqrt{d x + 1} - \sqrt{d x - 1}\right )}^{8} - 96 \, a d^{4}{\left (\sqrt{d x + 1} - \sqrt{d x - 1}\right )}^{4} - 96 \, c d^{2}{\left (\sqrt{d x + 1} - \sqrt{d x - 1}\right )}^{4} - 48 \, b d^{3}{\left (\sqrt{d x + 1} - \sqrt{d x - 1}\right )}^{2} - 128 \, a d^{4} - 192 \, c d^{2}\right )}}{{\left ({\left (\sqrt{d x + 1} - \sqrt{d x - 1}\right )}^{4} + 4\right )}^{3}}}{3 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)/(sqrt(d*x + 1)*sqrt(d*x - 1)*x^4),x, algorithm="giac")

[Out]

-1/3*(3*b*d^3*arctan(1/2*(sqrt(d*x + 1) - sqrt(d*x - 1))^2) + 2*(3*b*d^3*(sqrt(d
*x + 1) - sqrt(d*x - 1))^10 - 12*c*d^2*(sqrt(d*x + 1) - sqrt(d*x - 1))^8 - 96*a*
d^4*(sqrt(d*x + 1) - sqrt(d*x - 1))^4 - 96*c*d^2*(sqrt(d*x + 1) - sqrt(d*x - 1))
^4 - 48*b*d^3*(sqrt(d*x + 1) - sqrt(d*x - 1))^2 - 128*a*d^4 - 192*c*d^2)/((sqrt(
d*x + 1) - sqrt(d*x - 1))^4 + 4)^3)/d